Pancakes With A Problem! Great Theoretical Ideas In Computer Science Anupam Gupta CS Fall 2O05 Lecture 1 Aug 29 th, 2OO5 Aug 29 th, 2OO5 Carnegie Mellon University

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Pancakes With A Problem! Great Theoretical Ideas In Computer Science Anupam Gupta CS Fall 2O05 Lecture 1 Aug 29 th, 2OO5 Aug 29 th, 2OO5 Carnegie Mellon University

The chef at our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top, and so on, down to the largest at the bottom) I do this by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary.

Developing A Notation: Turning pancakes into numbers

How do we sort this stack? How many flips do we need?

4 Flips Are Sufficient

Algebraic Representation X = The smallest number of flips required to sort: ?  X  ? Upper Bound Lower Bound

Algebraic Representation X = The smallest number of flips required to sort: ?  X  4 Upper Bound Lower Bound

Better Upper/Lower Bound? ?  X  4

4 Flips are necessary in this case Flip 1 has to put 5 on bottom Flip 2 must bring 4 to top.

Upper Bound Lower Bound 4  X  4 X = 4

5 th Pancake Number The number of flips required to sort the worst case stack of 5 pancakes. ?  P 5  ? Upper Bound Lower Bound P 5 =

5 th Pancake Number The number of flips required to sort the worst case stack of 5 pancakes. 4  P 5  ? Upper Bound Lower Bound P 5 =

..... The 5 th Pancake Number: The MAX of the X’s X1X1 X2X2 X3X3 X 119 X

The 5 th Pancake Number: The MAX of the X’s X1X1 X2X2 X3X3 X 119 X

X1X1 X2X2 X3X3 X 119 X P 5 = MAX over s 2 stacks of 5 of MIN # of flips to sort s

P n = MAX over s 2 stacks of n pancakes of MIN # of flips to sort s P n = The number of flips required to sort the worst-case stack of n pancakes.

P n = MAX over s 2 stacks of n pancakes of MIN # of flips to sort s P n = The number of flips required to sort a worst-case stack of n pancakes.

Be Cool. Learn Math-speak. P n = The number of flips required to sort a worst-case stack of n pancakes.

What is P n for small n? Can you do n = 0, 1, 2, 3 ?

Initial Values Of P n n0123 PnPn 0013

P 3 = requires 3 Flips, hence P 3 ≥ 3. ANY stack of 3 can be done by getting the big one to the bottom ( ≤ 2 flips), and then using ≤ 1 extra flip to handle the top two. Hence, P 3 = 3.

n th Pancake Number The number of flips required to sort a worst case stack of n pancakes. ?  P n  ? Upper Bound Lower Bound P n =

Bracketing: What are the best lower and upper bounds that I can prove? ≤ f(x) ≤ [ ]

?  P n  ? Take a few minutes to try and prove upper and lower bounds on P n, for n > 3.

Bring biggest to top. Place it on bottom. Bring next largest to top. Place second from bottom. And so on… Bring-to-top Method

Upper Bound On P n : Bring To Top Method For n Pancakes If n=1, no work required - we are done! Otherwise, flip pancake n to top and then flip it to position n. Now use: Bring To Top Method For n-1 Pancakes Total Cost: at most 2(n-1) = 2n –2 flips.

Better Upper Bound On P n : Bring To Top Method For n Pancakes If n=2, at most one flip and we are done. Otherwise, flip pancake n to top and then flip it to position n. Now use: Bring To Top Method For n-1 Pancakes Total Cost: at most 2(n-2) + 1 = 2n –3 flips.

Bring to top not always optimal for a particular stack Bring-to–top takes 5 flips, but we can do in 4 flips

?  P n  2n-3 What other bounds can you prove on P n ?

Better Upper/Lower Bounds?

9 16 Suppose a stack S contains a pair of adjacent pancakes that will not be adjacent in the sorted stack. Any sequence of flips that sorts stack S must involve one flip that inserts the spatula between that pair and breaks them apart. Breaking Apart Argument

9 16 Suppose a stack S contains a pair of adjacent pancakes that will not be adjacent in the sorted stack. Any sequence of flips that sorts stack S must involve one flip that inserts the spatula between that pair and breaks them apart. Furthermore, this same principle is true of the “pair” formed by the bottom pancake of S and the plate. Breaking Apart Argument

n  P n n n-1 S Suppose n is even. S contains n pairs that will need to be broken apart during any sequence that sorts stack S.

n  P n Suppose n is even. S contains n pairs that will need to be broken apart during any sequence that sorts stack S S Detail: This construction only works when n>2

n  P n Suppose n is odd. S contains n pairs that will need to be broken apart during any sequence that sorts stack S n n-1 S

n  P n S Detail: This construction only works when n>3 Suppose n is odd. S contains n pairs that will need to be broken apart during any sequence that sorts stack S.

n  P n  2n – 3 for n ≥ 3 Bring To Top is within a factor of two of optimal!

Starting from ANY stack we can get to the sorted stack using no more than P n flips. n  P n  2n – 3 for n ≥ 3

From ANY stack to sorted stack in ≤ P n. Reverse the sequences we use to sort. From sorted stack to ANY stack in ≤ P n ? ((( )))

Hence, From ANY stack to ANY stack in ≤ 2P n. From ANY stack to sorted stack in ≤ P n. From sorted stack to ANY stack in ≤ P n ?

From ANY stack to ANY stack in ≤ 2P n. Can you find a faster way than 2P n flips to go from ANY to ANY? ((( )))

Any-to-Any? S: 4,3,5,1,2 T: 5,2,4,3,1

From ANY Stack S to ANY stack T in ≤ P n Rename the pancakes in S to be 1,2,3,..,n. Rewrite T using the new naming scheme that you used for S. T will be some list:  (1),  (2),..,  (n). The sequence of flips that brings the sorted stack to  (1),  (2),..,  (n) will bring S to T. S: 4,3,5,1,2 T: 5,2,4,3,1 1,2,3,4,53,5,1,2,4

The Known Pancake Numbers n PnPn

P 14 Is Unknown 14! Orderings of 14 pancakes. 14! = 87,178,291,200

Is This Really Computer Science?

Posed in Amer. Math. Monthly 82 (1) (1975), “Harry Dweighter” a.k.a. Jacob Goodman

(17/16)n  P n  (5n+5)/3 William Gates & Christos Papadimitriou Bounds For Sorting By Prefix Reversal. Discrete Mathematics, vol 27, pp 47-57, 1979.

(15/14)n  P n  (5n+5)/3 H. Heydari & H. I. Sudborough On the Diameter of the Pancake Network. Journal of Algorithms, vol 25, pp 67-94, 1997.

Permutation Any particular ordering of all n elements of an n element set S, is called a permutation on the set S. Example: S = {1, 2, 3, 4, 5} One possible permutation: *4*3*2*1 = 120 possible permutations on S

Permutation Any particular ordering of all n elements of an n element set S, is called a permutation on the set S. Each different stack of n pancakes is one of the permutations on [1..n].

Representing A Permutation We have many ways to specify a permutation on S. Here are two methods: 1)We list a sequence of all the elements of [1..n], each one written exactly once. Ex: )We give a function  on S such that  (1)  (2)  (3)..  (n) is a sequence that lists [1..n], each one exactly once. Ex:  (1)=6  (2)=4  (3) = 5  (4) = 2  (4) = 1  (6) =3

A Permutation is a NOUN An ordering S of a stack of pancakes is a permutation.

A Permutation is a NOUN. A permutation can also be a VERB. An ordering S of a stack of pancakes is a permutation. We can permute S to obtain a new stack S’. Permute also means to rearrange so as to obtain a permutation of the original.

Permute A Permutation. I start with a permutation S of pancakes. I continue to use a flip operation to permute my current permutation, so as to obtain the sorted permutation.

There are n! = 1*2*3*4*…*n permutations on n elements. Easy proof in the first counting lecture.

Pancake Network: Definition For n! Nodes For each node, assign it the name of one of the n! stacks of n pancakes. Put a wire between two nodes if they are one flip apart.

Network For n=

Network For n=4

Pancake Network: Message Routing Delay What is the maximum distance between two nodes in the network? PnPn

Pancake Network: Reliability If up to n-2 nodes get hit by lightning the network remains connected, even though each node is connected to only n-1 other nodes. The Pancake Network is optimally reliable for its number of edges and nodes.

Mutation Distance

High Level Point Computer Science is not merely about computers and programming, it is about mathematically modeling our world, and about finding better and better ways to solve problems. This lecture is a microcosm of this exercise.

One “Simple” Problem A host of problems and applications at the frontiers of science

Study Bee Please read the course document carefully. You must hand in a signed cheating policy page.

Study Bee Definitions of: nth pancake number lower bound upper bound permutation Proof of: ANY to ANY in ≤ P n Important Technique: Bracketing

References Bill Gates & Christos Papadimitriou: Bounds For Sorting By Prefix Reversal. Discrete Mathematics, vol 27, pp 47-57, H. Heydari & H. I. Sudborough: On the Diameter of he Pancake Network. Journal of Algorithms, vol 25, pp 67-94, 1997